Affine deformations of Minkowski spaces

Abstract

We investigate generalized Berwald spaces Bⁿ over Rⁿ (local theory). These are Finsler spaces admitting metric linear connections Γ* over TRn. (If Γ* is torsion free, then Bⁿ is a Berwald space.) An affine deformation is a regular linear transformation of each TRⁿ. This takes the indicatrices of a Minkowski space Mn into other indicatrices, and thus it leads to a new Finsler space. We prove that any Bn is the affine deformation of an Mn, and conversely. We show that any Bn can be represented by a pair (Vⁿ, Mⁿ) of a Riemannian and a Minkowski space. Several properties of Bn will be expressed by properties of V n or Mn. Also, the linear automorphisms of the indicatrices will be investigated.